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Full text of "Treatise On Analysis Vol-Ii"

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1.    Let G be a locally compact group containing a compact open subgroup H. For each
automorphism u of G, show that mod(^) is a rational number. (Observe that w(H) n H
is a subgroup of finite index in both H and #(H).) Show that the set of elements s e G
such that AG(,s) = 1 is an open subgroup of G, containing H.

2.    Let G be a compact group, /z a Haar measure on G, and u a (continuous) endomor-
phism of G such that u(G) is open in G and the kernel Gu = u~l(e) is a finite subgroup
of G.

(a)    Show that there exists a real number h(u) > 0 and an open neighborhood U
of e in G such that, for every open set V c U, the set u(V) is open in G and
/A((V)) = K)/i(V) (use (14.2.5)).

(b)    Show that h(u) Card(G/w(G))/Card(Gu). (Calculate /x(w(G)) in two different
ways, using (a) above and (14.4.2).)

3.    Let Q* be the set of nonzero rational numbers, endowed with the discrete topology.
On the locally compact space G = R xQ* a law of composition is defined by the
formula (x, r)(x', r') = (rx' + x, rr'). Show that with respect to this law of composition
G is a locally compact group which is locally isomorphic to the additive group R,
but not unimodular.

4.    (a)   Let G be a locally compact group and let % be a continuous homomorphism of
G into the multiplicative group C*. Show that if a complex measure v on G is such
that Y(S)I/ = %(s)v for all s e G, then v  ax ' ^ where /x is a left Haar measure on G,
and a is a complex constant.

(b) A complex measure v on G is said to be left quasi-invariant if ^(s)v is equivalent
to v (13.15.6) for each s e G. Show that v is left quasi-invariant if and only if v is a
measure equivalent (13.15.6) to a left Haar measure ^ on G. (Reduce to the case v ^> 0;
use the criterion (b') of (13.15.5), consider the double integral


where A is compact, / e ^T(G), and / J> 0, and use the theorem of Lebesgue-Fubini.)

5. (a) Let G be a locally compact group and let X, Y be two closed subgroups of G
such that X n Y= {e} and such that the set H = XY (i.e., the set of all xy, where
x e X and y e Y) contains a neighborhood of e in G. Show that Q is open in G and
that the mapping (x, y)\-+xy~l of X x Y onto O, is a homeomorphism (cf. (12.16.12),
by considering X X Y as acting on ) by the rule (x, y)  z = xzy~ 1).
(b) Let IJLG , fjLX , JW-Y be left Haar measures on G, X, Y, respectively, and let fi be the
restriction of JJLG to O. Show that, up to a constant factor, /x, is the image of
MX  (x'1 ' J^-Y) under the homeomorphism (x, y)\-+xy"~l of X x Y onto H, where x
denotes the restriction of AG to Y. Deduce that a real-valued function /defined on O
is jii-integrable if and only if the function (x9y)^-^f(xy)^G(y)^(y)""1 is (/XX/XY)-
integrable, and that we then have

= a
where a is a constant independent of/subset P (resp. C) of X is said to be a